please HELP me with this work

Question

Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe^3x + αx^2 if x ≥ 0 is continuous on the interval (−∞,∞).

(b)For which values of α and β is the function f(x) differentiable?

Answer 1

I'm unsure if the first equation is f(x) = αcos(x) + 2 or f(x) = αcos(x + 2). I'll assume the first.

You want to find values of α and β where the two functions have the same function value at x = 0 (the function is then continuous) and they also have the same slope at x = 0 (the function then has no sharp points). You will get two equations in two unknowns and will be able to then solve the system of equations.

1) Set αcos(x) + 2 equal to βe^3x + αx² and make x = 0 making the equation α + 2 = β

2) Take the derivatives and set them equal. Then solve the system to get α and β.

To double check your answers, plot the two functions and look at x = 0 to make sure it is a continuous and smooth transition.

Answer 2

I agree with the First step of your solution.

If I follow your instruction in 2nd step I get β = 0 which is a trivial solution and α = -2

With these solutions the functions are f(x) = -2cos x +2 for x < 2

and f(x) = -2x^2 for x ≥ 2 which is a parabola

William please clarify, and provide answer to (b) part.

Shailesh K.